Cauchy Problem for Fractional Diffusion-Wave Equations with Variable Coefficients

We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order $α\in (1,2)$ with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial variables. This equation describes the propagation of stress pulses in a viscoelastic medium. Its properties are intermediate between those of parabolic and hyperbolic equations. In this paper, we construct and investigate a fundamental solution of the Cauchy problem, prove existence and uniqueness theorems for such equations.

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